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Informatics and Mathematical Modelling / Intelligent Signal Processing Sparse Coding and Automatic Relevance Determination for Multi-way models Morten Mørup Joint work with DTU Informatics Intelligent Signal Processing Technical University of Denmark Sparse’09 8 April 2009 Lars Kai Hansen DTU Informatics Intelligent Signal Processing Technical University of Denmark 1 Informatics and Mathematical Modelling / Intelligent Signal Processing Vector: 1-way array/ 1st order tensor, Matrix: 2-way array/ 2nd order tensor, 3-way array/3rd order tensor Multi-way modeling has become an important tool for Unsupervised Learning and are in frequent use today in a variety of fields including Psychometrics Chemometrics NeuroImaging Textmining Signal Processing Sparse’09 8 April 2009 (Subject x Task x Time) (Sample x Emission x Absorption) (Channel x Time x Trial) (Term x Document x Time/User) (ICA, i.e. diagonalization of Cummulants) 2 Informatics and Mathematical Modelling / Intelligent Signal Processing Some Tensor algebra XI x J x K I x JK Matrix X(1) Matricizing X X(n) X(2) X(3) N-mode multiplication J x KI Matrix K x IJ Matrix i.e. for matrix multiplication CA=A x1C , ABT=A x2B Sparse’09 8 April 2009 3 Informatics and Mathematical Modelling / Intelligent Signal Processing Tensor Decomposition The two most commonly used tensor decomposition models are the CandeComp/PARAFAC (CP) model and the Tucker model Sparse’09 8 April 2009 4 Informatics and Mathematical Modelling / Intelligent Signal Processing CP-model unique Tucker model not unique What constitutes an adequate number of components, i.e. determining J for CP and J1,J2,…,JN for Tucker is an open problem, particularly difficult for the Tucker model as the number of components are specified for each modality separately Sparse’09 8 April 2009 5 Informatics and Mathematical Modelling / Intelligent Signal Processing Agenda To use sparse coding to Simplify the Tucker core forming a unique representation as well as enable interpolation between the Tucker (full core) and CP (diagonal core) model To use sparse coding to turn off excess components in the CP and Tucker model and thereby select the model order. To tune the pruning strength from data by Automatic Relevance Determination (ARD). Sparse’09 8 April 2009 6 Informatics and Mathematical Modelling / Intelligent Signal Processing Sparse Coding Nature, 1996 Bruno A. Olshausen David J. Field Preserve Information Preserve Sparsity (Simplicity) Tradeoff parameter Sparse’09 8 April 2009 7 Informatics and Mathematical Modelling / Intelligent Signal Processing Sparse Coding (cont.) Image patch 1 to N … Patch image Vectorize patches Sparse coding attempts to both learn dictionary and encoding at the same time (This problem is not convex!) Sparse’09 8 April 2009 8 Informatics and Mathematical Modelling / Intelligent Signal Processing Automatic Relevance Determination (ARD) Automatic Relevance Determination (ARD) is a hierarchical Bayesian approach widely used for model selection In ARD hyper-parameters explicitly represents the relevance of different features by defining the range of variation. (i.e., Range of variation0 Feature removed) Sparse’09 8 April 2009 9 Informatics and Mathematical Modelling / Intelligent Signal Processing A Bayesian formulation of the Lasso / BPD problem Sparse’09 8 April 2009 10 Informatics and Mathematical Modelling / Intelligent Signal Processing ARD in reality a L0-norm optimization scheme. As such ARD based on Laplace prior corresponds to L0-norm optimization by re-weighted L1-norm L0 norm by re-weighted L2 follows by imposing Gaussian prior instead of Laplace Sparse’09 8 April 2009 11 Informatics and Mathematical Modelling / Intelligent Signal Processing Sparse Tucker decomposition by ARD Brakes into standard Lasso/BPD sub-problems of the form Sparse’09 8 April 2009 12 Informatics and Mathematical Modelling / Intelligent Signal Processing Solving efficiently the Lasso/BPD sub-problems Notice, in general the alternating sub-problems for Tucker estimation have J<I! Sparse’09 8 April 2009 13 Informatics and Mathematical Modelling / Intelligent Signal Processing The Tucker ARD Algorithm Sparse’09 8 April 2009 14 Informatics and Mathematical Modelling / Intelligent Signal Processing Results Fluorescene data: emission x excitation x samples Tucker(10,10,10) models were fitted to the data, given are below the extracted cores Synthetic Tucker(3,4,5) Sparse’09 Tucker(3,6,4) 8 April 2009 Tucker(3,3,3) 15 Tucker(?,4,4) Tucker(4,4,4) Informatics and Mathematical Modelling / Intelligent Signal Processing Sparse Coding in fact 2-way case of Tucker model The presented ARD framework extracts a 20 component model when trained on the natural image data of (Olshausen and Field, Nature 1996) Image patch 1 to N … Patch image Vectorize patches Sparse’09 8 April 2009 16 Informatics and Mathematical Modelling / Intelligent Signal Processing Conclusion Imposing sparseness on the core enable to interpolate between Tucker and CP model ARD based on MAP estimation form a simple framework to tune the pruning in sparse coding models giving the model order. ARD framework especially useful for the Tucker model where the order is specified for each mode separately which makes exhaustive evaluation of all potential models expensive. ARD-estimation based on MAP is closely related to L0 norm estimation based on reweighted L1 and L2 norm. Thus, ARD form a principled framework for learning the sparsity pattern in CS. Sparse’09 8 April 2009 17 Informatics and Mathematical Modelling / Intelligent Signal Processing http://mmds.imm.dtu.dk Sparse’09 8 April 2009 18